Sheaves and Algebraic Geometry

Normality, regularity, smoothness, flatness, etale, unramified: big picture

How are these related? What are different ways of thinking about these (e.g. regularity vs. locally free sheaf of differentials)? How are these used (e.g. regular hypothesis is nice for Weil to Cartier divisor correspondence)?

For example, a normal (integral) scheme is regular in codimension one.

Vakil (see e.g. here) is a good self-contained, detailed reference for things like regular embeddings, left-exactness of the conormal exact sequence, Bertini's theorem, etc.

Normality is also used in Corollary 7.8 and 7.9 of Hartshorne, Chapter III.7 (on Serre duality), which includes as special case Remark 7.9.1 that the smooth hyperplane sections in Bertini's theorem are in fact connected, hence irreducible.

Affine, separated, (universally) closed, projective, proper

Examples and non-examples of each. Different ways of thinking about these. Connection to topology notions (contractible, Hausdorff, compact, proper). How are these used?

For affine and projective it's not clear whether best to discuss absolute or relative notion. For the rest, I guess they are all relative, but examples in absolute case can be helpful.

From a classical, qualitative intersection theory perspective, the reason projectivization is a good idea is that dimension theory works much better: in $\mathbb{P}^n$, two closed subvarieties, of codimension $a$ and $b$, must intersect if $a+b \leq n$. The argument is pretty nice: think about homogeneous polynomials on $\mathbb{A}^{n+1}$. Since the pullbacks intersect at the origin, and by affine dimension theory all components of the intersection have codimension at most $$a+b\leq n < n+1 = \dim \mathbb{A}^{n+1},$$ the component of zero must have dimension at least $(n+1) - n = 1$ (i.e. positive dimension), so back in the projective world, that means there must be a point.

Morphisms to $\mathbb{A}^d$ and $\mathbb{P}^d$, ampleness, and quasiprojective revisited

Compare:

• for a morphism $f\colon X \to \mathbb{A}^d$, we have $f^\ast \mathcal{O}_{ \mathbb{A}^d} = \mathcal{O}_X$, which (since $\mathbb{A}^d$ is affine with free ring of global sections) is the same as giving a list of $d$ global sections of the structure sheaf $\mathcal{O}_X$; whereas
• for a morphism $f\colon X \to \mathbb{P}^d$, we have $f^\ast (\mathcal{O}_{ \mathbb{P}^d}(1)) = \mathcal{L}$ for some invertible sheaf $\mathcal{L}$ on $X$ generated (on stalks) by a list of $d+1$ global sections. (It is still true that $f^\ast \mathcal{O}_{ \mathbb{P}^d} = \mathcal{O}_X$, as for any morphism, but it is easier to reason with $\mathcal{O}_{ \mathbb{P}^d}(1)$ than $\mathcal{O}_{ \mathbb{P}^d}$ here.)

In both cases, the concept of the pullback of a section can be used to give a uniform treatment (though with $\mathbb{A}^d$ this is arguably overcomplicating things): given $s\colon \mathcal{O}_Y \to \mathcal{F}$ on $Y$, the pullback under $\pi\colon X\to Y$ induces $\pi^\ast s\colon \mathcal{O}_X = \pi^\ast \mathcal{O}_Y \to \pi^\ast \mathcal{F}$. (After all, pullback is a functor between two categories of sheaves.)

Note: (Bi)rational maps $X\dashrightarrow \mathbb{P}^1$ ("meromorphic functions") arise naturally as ratios of sections of a line bundle $\mathcal{L}$ on $X$. Confusion with rational morphisms $X \to \mathbb{P}^1$ may arise due to results like MSE: Rational map is not defined on a subset of codimension $\geq2$ that let us extend past codimension one (at least if $X$ is smooth or normal or something), which is especially useful if $X$ is a smooth curve.

A decent treatment for maps to projective space, $\mathbb{P}^d$, is Qing Liu, Chapter 5, Proposition 1.31 and Remark 1.32.

A potentially confusing/enlightening example is:

• $\mathbb{A}^2 \to \mathbb{P}^3$ with "$(a,b)\mapsto [a:b:1:1]$" informally. Say $\mathbb{A}^2 = \operatorname{Spec} k[x,y]$.

• Formally, this is an open immersion factoring through $$\mathbb{A}^2 \to D_+(T_3) \cong \operatorname{Spec}(k[T_0,T_1,T_2,T_3]_{T_3,0}) \cong \operatorname{Spec}(k[T_0/T_3,T_1/T_3,T_2/T_3])$$ (the $0$ meaning degree zero as usual), corresponding to a map on sections $T_0/T_3\mapsto x$ and $T_1/T_3\mapsto y$ and $T_2/T_3\mapsto 1$. By niceness of pullbacks, we can compute $f^\ast \mathcal{O}_Y(1)$ on $V = D_+(T_3)$ (say) with $U = f^{-1}(V) = X$ to get that $f^\ast \mathcal{O}_Y(1)$ is the quasicoherent tilde (on $\mathbb{A}^2 = \operatorname{Spec} k[x,y]$, which is affine) of $$k[x,y]\otimes_{k[T_0/T_3,T_1/T_3,T_2/T_3]} T_3 \cdot k[T_0/T_3,T_1/T_3,T_2/T_3],$$ with $f^\ast T_0$ corresponding to the tensor $$1\otimes T_3\cdot (T_0/T_3) = x\otimes T_3$$ by the definition $T_0/T_3\mapsto x$. Similarly, $f^\ast T_1$ corresponds to $y\otimes T_3$, and $f^\ast T_2$ corresponds to $1\otimes T_3$. Of course, there is no harm in dropping the "$\otimes T_3$" if we identify $$k[x,y]\otimes_{k[T_0/T_3,T_1/T_3,T_2/T_3]} T_3 \cdot k[T_0/T_3,T_1/T_3,T_2/T_3] \cong k[x,y].$$

• Alternatively, we can choose $\mathcal{L}$ to be $\mathcal{O}_X$ (free of rank one, so certainly invertible) with $(s_0,s_1,s_2,s_3) = (x,y,1,1)$, which by the general theory (see e.g. Qing Liu reference above) defines a map $f\colon \mathbb{A}^2 \to \mathbb{P}^3$ with $s_i = f^\ast T_i$.

Note: To gain geometric intuition about hyperplane sections, degree of divisors (at least for curves, or more generally in intersection theory contexts), divisors of zeros (or poles, or... and connection to fibers of rational functions $X\dashrightarrow \mathbb{P}^1$), linear systems (e.g. all effective divisors equivalent to a given, possibly abstract, line bundle), linear equivalence (e.g. of effective Cartier divisors on a normal projective variety), ampleness (esp. for projective as opposed to just quasiprojective varieties), global generation (the condition for $X \to \mathbb{P}(V)$ to be well-defined as is), very ampleness, etc. maybe one should apply these concepts, e.g. dimension-counting moduli space of morphisms $C\to \mathbb{P}^1$ of given degree and genus, or Bertini's theorem, or intersection theory...

Global sections, distinguished (e.g. affine) sections, and Cech cohomology

"Local-to-global" arguments are key. Vaguely speaking, this includes gluing, patching, checking global exactness using stalks (or related sheaf-based ideas), and partitions of unity.

Note: Algebraically, we care about (global) sections on different sheaves (possibly on different related schemes) and how they are related to each other (e.g. questions of lifting sections along a sheaf surjection). The local-to-global gluing/patching arguments are encapsulated cleanly through sheaf cohomology, long exact sequence (and additivity of Euler characteristic), etc.

Note: The geometric significance of sheaf cohomology arguably lies in the fact that the global topology of $X$ (the underlying scheme) roughly determines the nontrivial-ness of the cohomology. (On affines, stuff is trivial, so the nontrivial-ness comes from gluing the affines together.) Conversely, cohomology $H^\ast$ of certain sheaves (or related invariants like Euler characteristic $\chi$) can encode geometric/topological information about $X$.

Partitions of unity can be used for the following (which I might write out if I have time):

• Proving localization defines a structure sheaf on $\operatorname{Spec}R$. More generally, proving that localization defines a sheaf $\tilde{M}$ on $\operatorname{Spec}R$ associated to any $R$-module $M$. Kedlaya calls this the "first fundamental theorem of affine schemes" in his intro notes on schemes (from MIT's 18.726 in Spring 2009).
• Proving that on an affine scheme $X = \operatorname{Spec}R$, taking global sections is exact (right-exact being what is not true in general) given $0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$ with $\mathcal{F}_1$ quasicoherent. And the related fact that the cohomology (Cech, say) $H^i(X, \mathcal{F}_1)$ vanishes for $i>0$. Kedlaya calls this the "fourth fundamental theorem of affine schemes" in his notes on cohomology of quasicoherent sheaves, though he takes a more abstract approach rather than using partitions of unity. For the latter with respect to a principal affine open cover of $X$, see the partitions of unity argument in Qing Liu, Chapter 5, Lemma 2.17. For the former, see Hartshorne Proposition II.5.6 or Qing Liu, Chapter 5, Proposition 1.8.

Remark: The "third fundamental theorem of affine schemes" is the classification of quasicoherent sheaves: see Kedlaya's intro notes on sheaves of modules, or another treatment in Qing Liu, Chapter 5, Proposition 1.6 and Theorem 1.7. I'm not sure what the "second fundamental theorem of affine schemes" is, but it might be Theorems 1 and 2 in Kedlaya's intro notes on morphisms of schemes.

Curve theory references

Standard references: Hartshorne, Qing Liu, Vakil. Stichtenoth is a good reference for the function field perspective, including a simple argument for asymptotic Riemann Roch.

See Roy Smith's answers on MO and MSE. Other users to read?

MO: Elementary Proof of Riemann-Roch for Compact Riemann Surfaces; see Roy Smith and David Speyer answers in particular. Terry Tao blog post.

Joe Harris course notes; see e.g. Atanas Atanasov notes.

Mohan: $\operatorname{Sym}^d C \to \operatorname{Pic}^d C$ (dimension $d$ mapping to dimension $g$) supposed to give heuristic RR dimension count (even with corrections?). Observe that $\operatorname{Sym}^d C$ corresponds to effective divisors, so a fiber of this map will contain all effective divisors of the form $(f)+D$ with $(f)$ principal and $D$ fixed in the divisor class $[D]\in \operatorname{Pic}^d C$ being considered (so $D$ need not be an effective representative, which indeed might not exist; the only effectiveness restriction is on the $(f)+D \geq 0$ points in the fiber, which will be empty if the class $[D]$ has no effective representative).

Also Mohan: some direct way of seeing $\chi(D+p) = \chi(D)+1$, with an underlying geometric result (gluing line bundles? or sections? or total spaces?) of independent (perhaps greater) interest: maybe inquire about details. What is the geometric intuition for Euler characteristics? See MSE: Meaning of holomorphic Euler characteristics?; MO: Why is the Euler characteristic of powers of a line bundle a polynomial in the power?; MO: Effect of tensor product on euler characteristic of line bundles; MO: Euler characteristic of a curve;

MO: Intuition behind Riemann's bilinear relations

MO: What does the Jacobian of a curve tell us about the curve?

MO: How do you see the genus of a curve, just looking at its function field?

MO: Elliptic Curves with CM and Class Field Theory

MSE: Constructing one-forms on a Riemann surface using the uniformization theorem and MO: The existence of meromorphic functions on Riemann surfaces. See Tsimerman generals.

Flatness to-do

• Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it?
• Use both algebra (e.g. number theory) and geometry intuition
• MO: Why are flat morphisms "flat?"

My version/interpretation of Critch's answer: In topology, if a map $X\to Y$ is continuous, limiting behavior in $X$ corresponds to limiting behavior in $Y$. This can be useful if, say, $Y$ is simpler than $X$, and/or if we want to say something about $X$ based on properties of $Y$, say by pulling back a closed or open set of interest from $Y$ to $X$; or say (on the algebraic/analytic side) by pulling back functions of interest from $Y$ to $X$, like idempotents or curves or something.

Now, say $Y = \operatorname{Spec}(A)$ is flat over $X = \operatorname{Spec}(B)$; then e.g. the injection $A\to \operatorname{Frac}(A)$ (generic analysis/functions on base $Y$) induces an injection $B\to B\otimes \operatorname{Frac}(A)$ (allowing generic fiber analysis on $X$; see Qing Liu Chapter 4, Lemma 3.7).

Or say $A,B$ are local and we want to do differential calculus (tangent/cotangent spaces, Taylor series, ramification, etale, smoothness, etc.). Then the short exact sequence $$0\to \mathfrak{m}_y^r \to A \to A/\mathfrak{m}_y^r\to 0$$ induces a short exact sequence $$0\to \mathfrak{m}_y^r\otimes_A B \to B \to B/ \mathfrak{m}_y^rB\to 0.$$ In particular, as Critch mentions, the usually-just-a-surjective-map $$\mathfrak{m}_y^r\otimes_A B \to \mathfrak{m}_y^rB$$ is an isomorphism, so $\mathfrak{m}_y^rB$ should have "no surprise relations" (my currently attempted/preferred wording: it is an ideal of $B$ "cut out and/or described in terms of $A$, at least as much as one can hope for").

Remark: For all of the above, I'm not sure how much intuition (geometric or otherwise) is missing if one pretends $B$ is free over $A$.

Sheaves of differentials

Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on).

Want to merge classical and modern perspectives on calculus, tangent/cotangent spaces, etc. See Liu and Vakil for examples, intuition, and pictures. At the beginning, Vakil's "Key fact 21.2.3." on generators and relations of module of differentials (see November 18, 2017 version) is quite nice, with a tight connection to the affine conormal exact sequence:

21.2.D. EXERCISE. Verify Key fact 21.2.3. (If you wish, use the affine conormal exact sequence, Theorem 21.2.12, to verify it; different people prefer to work through the theory in different orders. Just take care not to make circular arguments.)

Qing Liu Chapter 6, Proposition 1.8, relating differentials on towers of fiber spaces (i.e. "affine cotangent sequence" and "affine conormal sequence"). It's interesting that while differentials are (at least in some special cases of interest) easy to compute directly and explicitly by hand (at least intuitively, though checking the universal property may be boring), the proof goes more geometrically through derivations (Yoneda-esque dual perspective, using universal property of the module of relative differential forms). Reflect more deeply on this?

[EDIT after first reading: See also Vakil, "relative cotangent sequence, affine version" and "conormal exact sequence, affine version" for other discussion.]

Could perhaps be proved directly with more yoga, but the derivations sequence is much more transparent than the originals. I suppose it's not too honest to call derivations "purely geometric", but at least elements of the derivation sets are more or less tangent vectors, and the restriction and inducing of derivations can be understood geometrically in that way.

(Technical note: see the algebraic fact that let us conclude exactness from dual exactness: MSE: Dual sequence and its exactness; MSE: Verifying statement of exact sequences; MSE: Hom functors and exactness; MSE: Confusion about a proof: exactness of modules.)

Divisors, degree, and codimension one techniques

See e.g. Kedlaya's notes.

"Codimension one techniques" is kind of vague—but it could be useful to compile examples.

Blowups

MO: blowing up, -1 curves, effective and ample divisors

Understand how blowups are useful (e.g. allows us to work with divisors instead of points).

Cycles and intersection theory

Read Hartshorne?

See Vakil, Chapter 20 on intersection theory. It seems that invertible sheaves and effective Cartier divisors are the central concepts if one wants to work without assumptions (besides projective/proper) like smoothness or normality.

Visualize intersections of complex curves in four-dimensional real space? Should be transverse intersection, but can't currently wrap head around it.

Basics often left undefined or details unmentioned

See MSE: Does Hartshorne really not define things like the composition or restriction of morphisms of schemes?

In setting up closed immersions as in Qing Liu, useful to know that if $j\colon C\to X$ is closed immersion, and $\mathcal{F}$ a sheaf on $X$, then $\mathcal{F} \to j_\ast j^{-1} \mathcal{F}$ is always surjective, with stalks being killed away from $C$ (i.e. map on stalks is zero for $x\notin j(C)$; note that the complement of $j(C)$ in $X$ is open), and stalks isomorphic (if I'm not mistaken) for $x\in j(C)$. In particular, what is useful for closed immersions setup is that this is an isomorphism if $\mathcal{F}$ is supported on $C$. In this case, the isomorphic stalks at $x\in j(C)$ might be slightly easier to see too.

Pushforward and pullback (of sheaves?)

Examples (both algebraic and geometric/topological), contexts, applications, properties, how to think about these. See e.g. Wikipedia on inverse image functor, properties listed there.

Discuss how common things like adjunction and projection formula are not really as scary as they look.

Ringed spaces, morphisms, and immersions

Qing Liu, Section 2.2.2 "Ringed topological spaces" seems quite nice and concise. Maybe for newcomers could use more elaboration and examples to speed up learning process.

Spectrum of a ring: basics

Essentially, go back to big picture: why the Zariski topology? Why is this useful? Why study this and not something else? Connect this with classical examples, e.g. Hilbert's nullstellensatz.

For those with an arithmetic or algebraic bent, it could be more inspiring to bring sheaves into the picture earlier, not sure.

Sheaf mechanism

Why sheaves? Why is this natural? Basic examples of working with sheaves, e.g. stalk-to-local yoga, local-to-global gluing, partitions of unity, connection to classical study-module-by-localization ideas.

Basic non-relative scheme properties

Reduced, irreducible, integral

Connected, quasi-compact, locally Noetherian

Maybe dimension in nice cases?

Affine open subsets ("affine pieces glued together"), irreducible components, connected components

Could get side-tracked onto topology: points (fields), local/semi-local (number of closed points), etc.

Could get side-tracked into local vs. global notion discussion: e.g. Proj is arguably global notion, but being isomorphic to a Proj ("projective scheme" in Qing Liu terminology) is indeed a non-relative scheme property. Reduced, integral, and arguably irreducible can all be defined and/or checked on local rings.

Fiber products and their uses

Many key geometric notions can be captured through fiber products.

• base change, family, fiber, reduction
• restriction (e.g. $f^{-1}(V) \to V$ for $V$ open in $Y$), intersection (e.g. $U\cap V$ as fiber product)

Basic relative properties of schemes and morphisms

• Finite, affine
• Locally of finite type
• Quasi-compact (and how this is used, e.g. paired with (locally) Noetherian hypothesis)
• Dominant
• Closed
• Quasi-finite
• Generically finite

Generic points and specialization

How to think about this. Basically, it's a reinterpretation of Zariski topology.

How is this used? For example, an open property true for a single point in an irreducible component will automatically hold for the generic point of that component. Or how dominant morphism is reflected in generic point. Or how valuative criteria are formulated using generic point of a discrete valuation ring.

Common types of sheaves

• Quasicoherent, finitely generated, coherent, locally free, invertible
• Flasque resolution, e.g. $0 \to \mathcal{O}_X \to \mathcal{K}_X \to \mathcal{K}_X/ \mathcal{O}_X \to 0$; induced residue exact sequence (tensor with $\Omega_X$)
• $\mathcal{O}_X^\times$... exponential exact sequence, Picard group interpretation, $0 \to \mathcal{O}_X^\times \to \mathcal{K}_X^\times \to \mathcal{K}_X^\times/ \mathcal{O}_X^\times \to 0$, etc.

Why are these natural notions? Discuss equivalent definitions, e.g. locally $\mathcal{O}_X\vert_U$ presentation vs. locally of the form $\tilde{M}$. Discuss slogans like role of localization. And maybe more technical things like why noetherian and/or qcqs hypotheses can be useful in global gluing results.

Some tricky issues I don't understand, but I also don't think they are that important:

• Naturality of the boundary map for derived functors seems annoying to prove. See Wikipedia: "Derived functors and the long exact sequences are "natural" in several technical senses." The difficulty seems to be that a map of short exact sequences of chain complexes might not induce a commutative diagram on cohomology if we only assume it to commute up to chain homotopy; see MSE: Naturality of connecting morphisms found by googling "commutes up to chain homotopy". Eisenbud sort of sketchily leaves it to the reader; Weibel and Sharifi notes do this with a bit of work; not sure if this is worth looking at.
• Consider Cech-to-derived-functor-sheaf-cohomology map, Hartshorne Lemma III.4.4 (functoriality is believable, because we can probably just be lazy with commuting squares up to homotopy... but I don't easily see the naturality of the boundary map in long exact sequences). Compare annoying Hartshorne Theorem III.4.5 ($H^1$ on affines vanish (or just the fact that global sections functor is exact on affines when the leftmost term is quasicoherent) + Leray acyclicity theorem) and Exercise III.4.4 with hands-on Qing Liu Chapter 5 Proposition 2.15. It seems like the latter lets us work with Cech cohomology for all practical purposes (cf. Vakil treatment) without thinking about derived functor ("correct") approach at all. The comparison is comforting to know but maybe not that important in the end.
• Why do spectral sequences help us organize the comparison of cohomologies? See Wikipedia: Čech-to-derived functor spectral sequence.

Riemann-Roch over arbitrary fields

See MO: Is there a Riemann-Roch for smooth projective curves over an arbitrary field? or a book on function fields, like Stichtenoth, Algebraic Function Fields and Codes.

Connection to Tate's thesis & Poisson summation

See any good reference on Tate's thesis. See e.g. MSE: Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves? for an immediate online answer.

Benson Farb, Representation Stability

Matthew Baker, Hodge theory in combinatorics

Vector bundles vs. locally free sheaves

See Martin Brandenburg's answer to MSE: Equivalence of categories of vector bundles and locally free sheaves.

On the topic of monomorphisms of schemes, see MO: What are the monomorphisms in the category of schemes?.

Try out some confusing examples like $\operatorname{Spec}( \mathbb{Q}) \to \operatorname{Spec}( \mathbb{Z})$, using the characterization of maps from $X$ to an affine, and also fact that $\mathbb{Z} \to \mathbb{Q}$ is epimorphism in category of rings.